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                         *  F  E  A  T  U  R  E  S  *

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                       Chaos Reigns in a wonderful world

    Electronics  has played an important  role  in  one  of the  most  profound
and exciting developments of the 1980s,  perhaps of the  century - a new branch
of science known as chaos.
    Chaos  is  seen  by  some  as  rivalling  in  importance  such  fundamental
branches of science as quantum physics.
    It  consists  of new mathematical ways of  describing  a  huge  variety  of
phenomena  and objects that all  exhibit   irregular,  unstable patterns - from
clouds to rivers.
    With  conventional  mathematics  and  physics,  it  was  not  possible   to
describe  or predict the  formation  and   development   patterns  within  such
systems.   They  were  seen  as  randomly  erratic;  chaotic, infact.
    The patterns in a fast flowing river,   for example,  have been  until  now
literally  indescribable  by   conventional   techniques.    Such  'non  linear
systems' could not be modelled mathematically.
    The  new  science of chaos is  now  providing theories that  can  deal with
non linearity,  showing that  within  apparent  randomness   there is order and
pattern,  of a kind.   A universal constant of  chaos  has been discovered that
describes the emergence of  these   irregular   but patterned phenomena in many
different areas.   But  it  has  also   been  shown that chaotic  systems  will
always  be  inherently unpredictable.
    Chaos is founded on abstruse esoteric mathematics, but despite  that  there
have already been practical   applications,   such   as  improved  designs  for
machines to treat  heart  attack  victims.    Even  something  as  practical as
the  analysis  of  a  country's  economic   performance may well be improved by
the  techniques  of  chaos.
    Computers, and therefore the electronics  that  built them, have  played  a
crucial  part  in  the   emergence   of   chaos,   enabling  mathematicians and
physicists literally to see on a terminal  the  behaviour of chaotic systems.
    But  electronics  has played  a  more  central  role   than  merely  making
high  speed  digital computers  possible.   Some  of  the  foundation  work for
chaos was done by Stephen Smale,  a  leading  mathematician from the University
of California at Berkley.
    Smale's  work  was essentially   mathematical,   concerning  the  behaviour
of  all dynamical  systems.   But  he  was  led   to   it   by  considering the
behaviour of oscillating electronic circuits.
    These  circuits had been studied as  far  back as the  1920s  by  Balthasar
van der Pol, a Dutch electrical engineer.   He  analysed  the  behaviour  of an
electronic valve fed  with  varying  signal   levels.    When  van  der Pol did
the original  work,  he  had   no   oscilloscope   to  monitor the effects, and
instead  listened  to  alterations in the tone of a telephone ring.
    As  the  input varied, the tone  would  change in  a  series  of  frequency
jumps, locking into a particular  frequency  for  a   time  before  jumping  to
the  next.   But   sometimes    irregularities   occurred   before  the  jumps,
which  van  der   Pol   noticed   but   discounted.   These irregularities were
chaotic phenomena.
    The other major contribution of  electronics  to  the development  of chaos
came appropriately from Japan.
    Working  at the Department of Electrical Engineering at  Kyoto  University,
Yoshisuke  Ueda  published  a  paper   that   was    a  pioneering step towards
the science of chaos.
    It concerns random oscillations in  a series resonance  circuit  containing
a  saturable  inductor  under  the  impression  of  a  sinosoidal voltage.
    Ueda   introduced    his    paper    thus:    "In    physical    phenomena,
uncertainties  lie  between causes and effects.   When  uncertain  factors  are
small,  their  effects may   be   neglected   in   most  physical  systems  and
the  phenomena   under   consideration   are   treated  as  deterministic ones.
Whereas in non linear systems   on   some  conditions,  however small uncertain
factors may  be,  they  sometimes cause global changes."
    This summarises one of  the  central  principals  of  chaos,  known  as the
butterfly effect.   This  states  that  no  matter  how  small   a   degree  of
uncertainty exists about the state of   a  non  linear  system, the uncertainty
can be felt rapidly.
    A vivid way of illustrating  this  is  to  say  that  a single  flap  of  a
butterfly's wing in North America will  affect  the  state  of  the  weather in
China a few days later.  The  butterfly  effect  is  known  more technically as
'sensitivity to  initial  conditions',  and  this  sensitivity is infinite.  In
describing  the  initial  conditions of a non-linear dynamic system , no matter
what degree  of accuracy is achieved,  there will always be uncertainties.   No
matter how small they are, they will in a relatively short period  of time come
to affect the system noticeably.
    This  means that there  is  an  inherent  unpredictability  to  non  linear
systems, such as the weather, that  cannot  be  removed  ,  no  matter how much
knowledge about it grows.
    Ueda's work also concerned entities  called strange attractors,  which  lie
at  the  heart of the  mathematics   of  chaos.   They  describe  patterns that
emerge in chaotic systems, which  are  in  one sense stable in that they are of
similar types.  But they are  never repeated identically.
    For example, a pendulum  swinging  through  a  full  circle  driven  by  an
energetic  kick  at  regular  intervals   will   display   chaotic   behaviour,
tracking out a series of orbits that  are similar  but  which never revert back
precisely to a previous orbit.
    Ueda  demonstrated how the same  thing  happens with  electronic  circuits:
"The phenomenon should be  called  turbulence  in  electric  circuits.  Results
obtained disclose an important  feature  of   non   linear phenomena in general
physical systems."
    His paper shows that  the  randomness  displayed  by  the   circuits  is an
inherent feature of the system as a whole.
    Another  name  for  strange  attractors  is   fractal   attractors.   These
were  discovered  by Beniot Mandelbrot,   an  IBM  computer  scientist.    They
are  mathematical  entities   which    generate   infinitely  complex patterns,
repeating for ever according  to  a  recognisable theme but which can be proved
will never revert back  to any previously traced pattern.
    Again, electronics played a role in  Mandelbrot's  work.  He was  led to it
by work on intermittent noise in digital  communication  systems.
    Mandelbrot has been a pioneering figure  in the development  of  chaos  and
was helped in this by the fact that  he had  access  to  powerful  computers at
IBM in the late 1970s, machines  that   most   mathematicians did not then have
available to them.


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